This invention relates to uses of GPS data applied to direction and timing, and in particular to a presentation or display of such information.
As used herein a solar timer is a term referring to a package of functions incorporated in an application program comprising the present invention. It may be used in the form of a trademark SolarTimer to refer to a consumer entertainment tool made up of the package of these functions. Thus SolarTimer is a trademark of the assignee or licensee of the present invention identifying the source of the solar timer computer application package.
By way of background, the following information is provided as a tutorial on magnetic compass technology. Also following is a tutorial on GPS technology and a brief explanation of atmospheric considerations.
Magnetic Compass Basics and Earth's Magnetic Field
A conventional magnetic compass is used to set a constant direction for navigation. It is very cheap, lightweight and simple, and it can be made very small and it consumes no power. However, it has shortcomings related to its simplicity and the terrestrial environment. To understand the shortcoming issues of the magnetic compass it is useful to start from the basics of Earth's magnetic field [10, referring to cited references herein].
Earth has a magnetic field like a bar magnet which its north pole pointing towards to the Earth's magnetic north pole, a point on the northern hemisphere at which Earth's magnetic field points vertically downwards. Although it is in the northern hemisphere, by the direction of the magnetic field lines it is a “Magnetic South” pole. To be precise the “North Magnetic Pole” is in northern Canada near Ellesmere Island at 81.3° N and 110.8° W. This coordinate of the magnetic north pole is obtained from the geological survey of Canada done in 2001. The location of the Earth's magnetic north and south poles does not remain stationary; they change location over time by a small amount. It was estimated to be at 82.7° N and 114.4° W in 2005. In 2009 it was moving towards Russia at 55-60 km per year. As can be seen in FIG. 1, which is a graph of the movement of the Earth's rotational poles for the years 2001 to 2006, and showing the mean pole location for the years 1900 to 2000, the yearly change is relatively small when observed from a great distance, but it is large enough to cause navigation-related problems when traveling or surveying close or above latitudes close to the magnetic poles. (Units are milliarcseconds. This image is courtesy IERS Earth Orientation Center.) Therefore for centuries magnetic north was used successfully as a reference direction for magnetic compass usage.
There is also a “South Magnetic Pole” and due to the unsymmetrical nature of the Earth's magnetic field, it is not at the symmetrical position of North Magnetic Pole relative to the center of the Earth. So a line drawn between the North and South magnetic poles misses the center of the Earth by approximately 530 km. Both of the magnetic poles are also known as “Magnetic Dip Poles” due to the “dip” of the magnetic field lines at these points, which is also observed visually as the dip of the compass needle at these poles. South Magnetic Pole also changes with time. As an example it was at 64.6° 5 and 138.5° E in 1998 and it was estimated to be at 64.4° 5 and 137.3° E in 2010.
In addition, Earth's magnetic field strength and direction varies with location, meaning that a compass needle does not point exactly at Magnetic North everywhere on Earth. This has many reasons, but one of the main reasons is the non-uniform magnetic material composition of the Earths crust combined with the non-uniformity of the molten circulation currents in magma flow in the Earth's outer core, which is the basic source of Earth's magnetism.
Consider the effects of the non-uniform magnetic material composition in the Earth's outer crust which is solid. If there is a large iron or nickel ore deposit with magnetic properties in a localized area in the Earth's crust, one should expect some inaccuracies in the magnetic compass readings at those locations. Basalt is an iron-rich volcanic rock that covers the ocean floor and contains a strongly magnetic material called magnetite. It locally distorts the magnetic field lines of the Earth, which results in wrong heading and direction information to nearby compasses. This was discovered in the late 18th century by Icelandic mariners. Presence of the Earth's magnetic field gives the basalt measurable magnetic properties, which these magnetic data provided as another means to study the deep ocean floor. The changes in the Earth's magnetic field are recorded in the lava flow when it cools.
The recorded basalt magnetic data shows that Earth's magnetic field not only changed by a small amount, but actually reversed with an average period of 200,000 to 300,000 years in the past! However the last magnetic pole reversal event, which is also called Brunhes-Matuyama reversal, is calculated that it occurred approximately 780,000 years ago.
All these observable effects causing the compass needle not to point always towards the Magnetic North Pole everywhere on Earth is known as Magnetic Declination, and it is a very important piece of information in navigation [7-9]. The data is given in form of maps with contour lines of zero and equal declination that are called agonic and isogonic lines respectively. The magnetic field of Earth is measured at a very large number of sampling points on Earth by government organizations over time, because it varies considerably with time as well. Unfortunately variation does not have as long a period of time as the pole reversal. Therefore, updates are published at least twice a year. For example as of March 2010 in the San Francisco area, the magnetic north is about 14.3 degrees east of the true north, which is the geographical north, with the difference decreasing by about 6 minutes of arc per year.
Another good example to illustrate the importance of the Magnetic Declination can be found along the eastern seaboard of United States. The declination varies from 20 degrees west in Maine to zero degrees in Florida and 10 degrees east in Texas! This range of variation cannot be explained by the geographical location difference between the magnetic and true north. Thus a magnetic compass with a map without an updated Magnetic Declination map is insufficient for accurate navigation.
The magnetic field strength of the Earth was initially measured by Gauss in 1835 and has been measured periodically ever since by many means, including satellites such as Magsat and Orsted using very accurate 3-axis vector magnetometers [16]. The strength of the flux density at the Earth's surface ranges from less than 30 microteslas (0.3 gauss) in an area including most of South America and South Africa to over 60 microteslas (0.6 gauss) around the magnetic poles. The average flux density in the Earth's outer core is calculated to be 25 gauss, about 50 times stronger than the magnetic field at the surface. The flux density at the surface of the Earth is relatively small compared to any man-made permanent magnet in the close proximity of a magnetic compass. Therefore presence of a magnet or a magnetic material which has high magnetic permeability, μr, in close proximity to the magnetic compass will perturb the Earth's magnetic field locally resulting in incorrect direction finding [21,22]. As an example, silicon steel, which is used for transformers and electrical machinery such as electric motors, generators and relays, has magnetic permeability in the order of 40,000, iron and nickel have magnetic permeabilities in the order of 5,000 and 600 respectively. Presence of materials with such high permeabilities in large quantities will change the Earth's magnetic field locally as a function of their shape and their orientation with Earth's magnetic field. This property was used in submerged submarine detection and in magnetic mines since World War II. Even low flying aircraft with suitable instruments can detect the anomalies in the Earths magnetic field caused by submerged submarines. Therefore to have a compass that works correctly in the proximity of magnetic or ferro-magnetic material requires complex and costly calibration. This is done for all the navigation-grade compasses, making the compass large and heavy.
Other sources of spurious magnetic fields that disrupt a compass include any permanent magnet close to a magnetic compass and electric currents in general: Electric current produces a magnetic field [21,22]. Electric currents can be man made due to electric currents in electronic equipment nearby, due to power distribution networks or they can be natural, originating from space. Presence of a small electric current in the proximity of a compass is enough to perturb the Earth's magnetic field resulting in wrong reference direction finding. So, for accurate direction finding with a magnetic compass, all the electronic equipment should be kept away.
When a charged particle with a velocity enters an electromagnetic field a force called Lorentz force will be generated and which will act on the particle, resulting change its trajectory [21,22]. The magnetic component of the Lorentz force is determined by the sign of the charge of the charged particle and its vector product (curl) of the velocity vector v and the magnetic field flux density vector B. The presence of the Earth's magnetic field protects the Earth by deflecting most of the high energy bombardment of charged particles mainly originating from the sun due to this Lorentz force. Some of the charged particles are trapped in the Van Allen radiation belt that encircles the Earth. A smaller number of high energy charged particles manage to escape and interact with the gases in the upper atmosphere and ionosphere in the auroral zones creating beautiful bright auroras visible from the Earth's surface. These naturally occurring currents mainly generated in the upper atmosphere near the magnetic poles will perturb the compass reading as well.
For all these reasons there was a historic need for another method of reference direction setting in navigation. Gyro-compass based instruments historically answered this void in navigation very successfully. However, gyro-compasses are heavy, bulky and require very precise mechanics to manufacture, and they consume a large amount of power due to high rotational speeds required to operate accurately. They also have issues when used at very high latitudes like the magnetic compass. A recent design modification to the conventional gyro-compass is very accurate three axes accelerometers and laser gyros. However, these are expensive, require settings beforehand and are not available to everyone.
GPS Basics, Present Capabilities and Brief History
The Global Positioning System (GPS) is a navigation system that provides very accurate position, navigation and timing information any time and any place on Earth. By updating the position over time the system can also provide speed and directional information. Using information gathered from man-made satellites orbiting the Earth providing timing information with onboard extremely accurate atomic clocks, position can be determined to varying degrees of accuracy through calculations. Those calculations also show that if an observer on Earth can have a minimum of four satellites electronically visible, then by knowing their distances to the observer, location and altitude can be calculated anywhere on Earth. Excellent reading materials exist on the subject [1, 2], but it is useful to review some of the highlights of the GPS technology. Understanding requires expertise in very wide and seemingly unrelated areas, from surveying to satellite technology, from electronics to propagation and communication theory, from motion of objects in orbits to relativity theory and atomic clocks, and all are combined with complex calculations.
The accuracy of predicting the observer's location depends on many variables but most important is the knowledge of the distance to a minimum of four satellites as accurate as possible. This is achieved with very precise time information obtained from the atomic clocks in the GPS satellites that is sent along with the radio signal.
I. Smith's patent filed in 1964 describes a satellite system that would emit time codes and radio waves that would be received on Earth as time delayed transmissions creating hyperbolic lines of position. Several years later another patent filed by R. Easton refined the concept of comparing the phase from two or more satellites. In 1972 C. Counselman along with his colleagues in Massachusetts Institute of Technology (MIT) reported on the first use of interferometry to track the Apollo 16 Lunar Rover Module. The technique was applied in the development of the first geodetic GPS receiver and corresponding to differencing pseudo-ranges measured from two receivers to one satellite. The present use of the GPS carrier phase to make millimeter vector measurements dates to Very Long Baseline Interferometry (VLBI) work performed between 1976 and 1978. In 1986, B. Remondi first demonstrated that sub-centimeter vector accuracies could be obtained between a pair of GPS survey instruments with as little as a few seconds of data collection. B. Remondi later developed another survey technique which is called pseudo-kinematics, also known as intermittent static, snapshot static or reoccupation, which yields similar sub-centimeter accuracies. In this technique a pair of receivers occupies a pair of points for two brief periods that are separated in time. An Interferometric technology for codeless pseudo-ranging was developed by P. Mac Doran in Jet Propulsion Laboratory (JPL). With the Interferometric and VLBI techniques yielded the first portable codeless GPS receiver that could measure short baselines to millimeter accuracy and long baselines to one part per million. The codeless portable receiver development, trade-named Macrometer Interferometric Surveyor was demonstrated by the U.S. Federal Geodetic Control Committee. In 1981 U.S. National Geodetic Survey (NGS) and U.S. Geological Survey (USGS) developed specifications for portable dual frequency code correlating receivers that could be used for precise surveying and post processing. Texas Instruments was awarded the contract and produced the TI-4100 receiver. NGS geodesists C. Goad and B. Remondi developed the software to process its carrier phase data interferometrically as previously used by the MIT group.
In 1985 C/A code receivers started to output the carrier phase, and the first of these receivers was trade-named Trimble 400S and used vector computation software. Goad set the standard of the future software developers. At that time it required the processing power of a laptop computer to do the required computations.
For positioning any where on Earth, any time, more than four satellites are needed. The current GPS system consists of 24 evenly placed satellites in circular 12 hour orbits inclined 55 degrees to the equatorial plane forming a constellation. Currently there are more than 31 GPS satellites, with some used for back-up. They are referred to as NAVSTAR (Navigation Satellite Timing and Ranging) satellites. This near-circular orbit gives 20,200 km of altitude above the Earth. This configuration of satellites provides electronic visibility for minimum of 4 to 8 satellites with higher than 15 degrees of elevation from the horizon anywhere, anytime on Earth. For elevation angle of 10 degrees, or occasionally 10 degrees and 5 degrees of elevation, there will be 12 satellites electronically visible. In brief periods of time with an elevation angle of 10 degrees there can be up to 10 GPS satellites from which the most accurate positioning information can be obtained. Having more satellites electronically visible at higher elevation angles provides a means of making corrections in the calculations for much more accurate positioning.
Satellites basically provide a platform for radio transceivers, atomic clocks, computers and auxiliary equipment. There are six classes of satellites which are named Block 1, II etc., ranging from 845 kg to 2,000 kg, 5 meters across with two rubidium and two cesium atomic clocks with long-term frequency stability of a few parts in 10−13 and 10−14 per day. These atomic clocks almost synchronize everything, including the fundamental L-band frequency of 10.23 MHz. Coherently derived from this fundamental L-band signal are the L1 and L2 carriers which are at 1575.42 MHz and 1227.60 MHz respectively. The center frequencies of both bands are integer multiples of the L-band frequency of 10.23 MHz. The transmitter RF output power is 50 Watts or less.
Each satellite broadcasts two different direct-sequence spread spectrum signals with many lobes having a total bandwidth of 20 MHz. Having two signals is essential for eliminating the major source of error, which comes from the ionospheric refraction. The pseudoranges that are derived from measured travel times of the signal from each satellite to the receiver use two pseudorandom noise (PRN) codes that are modulated onto the two base carriers. PRN sequences are often called chips, and they do not carry data.
The first code is C/A-code, designated as the Standard Positioning Service (SPS) or known as coarse code is in the L1 band. The C/A-code uses 10.23 MHz chip rate. The main lobe of the C/A-code has a bandwidth of 2 MHz and occupies the entire 20 MHz bandwidth. The second code is the P-code (Precision-code), which is reserved for military and other authorized users, is broadcast in both L1 and L2 bands. The P-code is much more difficult to detect and uses a spreading code that only repeats at 1 week intervals and is encrypted. Its main lobe occupies the entire 20 MHz bandwidth due to a different chip rate, which is 10 times higher compared to the C/A-code. The signal strength in the L2 bands is half the strength compared to the L1 band. The outlying lobes of the P-code are truncated so that the entire GPS broadcast fits into its 20 MHz allocated bandwidth. While the P-code provide precise navigation information, by use of computational techniques, the C/A-code can be used to calculate reasonably accurate positioning information as explained below.
In addition to these codes a data message is modulated on the carriers consisting of status information, satellite clock bias, ephemerides and almanac data. The ephemerides data contains satellite ID number, current GPS week, ephemerides reference epoch, square root of semi-major axis, eccentricity, mean anomaly at reference epoch, argument of perigee, inclination, longitude of the node at weekly epoch, mean motion difference, rate of inclination angle, rate of node's right ascension correction coefficients, satellite clock reference epoch, satellite clock offset, satellite clock drift and satellite clock frequency drift. As can be seen the data is very comprehensive and using the data one can determine the satellite orbits and more.
Having a 50 Watts of RF transmit power and a distance of 20,200 km from the Earth surface going through the full thickness of ionosphere and troposphere results in a very low levels of received power on the surface of the Earth. The received power varies, but typically is around −130 dBm at the antenna of the receiver. For the primary lobe of the C/A-code which has a bandwidth of 2 MHz and for an antenna temperature of 290K the noise power is −111 dBm, approximately 20 dB lower than the noise floor, giving signal-to-noise ratio (SNR) at the antenna to be about −19 dB. Once the signal from a given satellite is correlated with the PRBS code, the bandwidth is reduced to only 100 Hz! So with a noiseless ideal receiver, the post-correlation SNR would be 24 dB.
To extract information from this level of very low power RF signals requires very high quality RF front ends and antenna designs in addition to sophisticated DSP and circuit techniques.
In commercial grade GPS, the accuracy in determining the location using only the C/A code is in the order of 10 meters. This accuracy can vary due to many factors such as terrain, loss of radio visibility to some satellites due to blocking objects such as buildings, multi-path effects and the like. More accurate GPS systems are available for civilian services by several means. Very precise location information is also provided by the introduction of WAAS or Wide Area Augmented System in 2004. GPS systems that utilize WAAS usually have accuracy of (1 to 3) meters. GPS systems can be even made more accurate by using subscription-based satellite correction or Real Time Kinematic local correction base stations to provide corrections to within 20 cm or about 9 inches for civilian use. (Military grade accuracy is better than these numbers.)
In a dedicated navigation or military grade receiver the designer has the luxury of design flexibility in a robust GPS reception system. The design of GPS system on a cell phone platform in an add-on application brings several limitations on basically everything from the antenna to the low noise amplifier (LNA). In addition to weight and size constraints there is also a radio frequency (RF) amplifier to perform its main function, which is communication, with an output power in the order of 900 mW, a frequency right next to the GPS receiver in frequency ranges of 800 MHz to 2.4 GHz, and thus a source of potential interference or desensing. Despite all of these disadvantages the resulting GPS capability provided in a smart phone today is surprisingly accurate. A typical smart phone or tablet computer can provide position information with an accuracy of 20 meters most of the time, anywhere on Earth and. Pushing it further probably is not needed for nonmilitary use.
As explained GPS is basically is a very sophisticated radio navigation tool. The earlier work on radio navigation goes back to at least the 1930's. Pioneered by German engineers in 1930's with the development of “Beams” or known as “Lorentz” system to guide the pilots for landings at night or poor visibility conditions, two highly directional radio transmitters are set side by side and transmit parallel radio signals. One signal consists of series of pulses in the form of a beep or dot. The other signal consists of a dash or a longer pulse. This distinct difference between the left and right signals gives the pilot a capability to fly the beam. In World War II both parties developed more advanced techniques to fly and navigate at night or under poor weather conditions with later work resulting in a system called HIRAN (High RAnge Navigation) using arcs of trilateration to position aircraft.
In 1946 in Finland an optical method based in principal on the stellar triangulation method was put in use very successfully. This required clear sky with a minimum of 4,000 km of separation between two observing sites. The equipment was large and expensive but it was successfully used for defining the European base line from Tromso in Norway to Catania in Sicily.
In the 1950's Dr. Ivan Getting made a breakthrough in radio navigation by developing the first “three dimensional position finding system based on the time of arrival” of radio signals that became the basis of GPS navigation. Conceptual catalyst for the satellite component of GPS was made by the launch of the first-man made satellite Sputnik in 1957 by the Soviet Union. Researchers in MIT realized that they could track Sputnik's orbit with the use of Doppler shift. This knowledge coupled with the ability to compute satellite ephemerides according to Kepler's laws led to the present capability of instantaneously determining precise position anywhere in the world.
The immediate predecessor of today's GPS system is the US Navy Satellite System (NNSS), also called as the TRANSIT system. The system had 6 satellites orbiting at altitudes of 1,100 km with nearly circular polar orbits.
Dr. Ivan Getting and Bradford Parkinson and MIT began development of the GPS in use today as a project for the US Department of Defense in 1973. Transmission was tested in 1977 and the first satellite launch was in 1978. The GPS satellite network and navigation system is owned by the US Government funded by the US Department of Defense developed maintained and operated by the US Air Force. It was declassified and made public in 1983 by President Ronald Reagan. It is free and a great service for all, provided by the US Government and it has had a great impact in navigation and the improvement of the air and sea travel safety since its implementation.
Computing Capabilities and Accuracy
There are many sites on the Earth's surface where the sun's elevation and azimuth angle calculations are done and published with accurate results, but mostly in the range of 60 degrees north latitude to 60 degrees south latitude. The general concepts qualitatively are explained very clearly in reference [19]. The reason for this is not because of lack of interest, or small population densities at higher or lower latitudes close to the poles, it is due to the mathematical difficulties in the calculations.
Calculation of the sun's elevation and azimuth angles as a function of time and date at higher and lower latitudes close to the poles becomes very complex. The basic reason of the complexity comes from the day and night at those higher latitudes can last more than 24 hours, even several months, depending on the date. The boundary for these regions is given by the Arctic Circle's definition. Arctic Circle is defined as the latitude which at least for one day a year the sun will be visible for 24 hour period [14]. The majority of the sun tracking algorithms and formulas take the day as 24 hours. However above the Arctic Circle in northern hemisphere and below the Antarctic Circle in the southern hemisphere, this assumption is invalid, and so the sun tracking algorithms and formulas with that assumption fail [2]. The position of the Arctic Circle changes with time by small amounts. For Epoch 2011 it is at latitude 66 Degree 33′44″ (66.5622 Degree) North. The Antarctic Circle is at the symmetric position of the Arctic Circle with respect to the Equator which is at 66 Degrees 33′44″ (66.5622 Degree) South.
As of Aug. 26, 2011 the sun elevation and azimuth angle calculation results of the present invention as hereinafter explained matches the NREL (National Renewable Energy Laboratory) results “exactly” below 68 Degree North or above 68 Degree South latitudes [11]. In the NREL Solar Positioning Algorithm (SPA) report [12] the claim is ±0.0003 degrees accuracy “where ever they can give results.” As of Aug. 26, 2011 NREL did not produce any data above 68 Degree North or below 68 Degree South latitudes. It is inferred that the program crashes for day lengths exceeding the 24 hours. Since the accuracy related claim given in the SPA report is ±0.0003 degrees accuracy “where ever they can give results”, this does not violate their claim, although it is incomplete.
Again, as of Aug. 26, 2011 the sunrise and sunset results produced by the present invention are within a second of NOAA (National Oceanic and Atmospheric Administration) results below 78 Degree North or above 78 Degree South latitudes [13]. For below 78 Degree South or above 78 Degree North latitudes NOAA gives the sunrise and sunset dates but with different results compared to observed and recorded dates as given in [15].
The present invention matches the historic data as published almost exactly at those extreme latitudes. The technical challenge is to predict the sun's elevation angle above the Arctic Circle over a period of a year, not merely on a selected convenient day. US Naval Observatory [15] and current invention results match extremely well in any latitude at any date, and both are all in line with historical measured data. This type of correlation with official USNO numbers over an entire year period, especially in the extreme latitude is considered remarkable. Finding this type of correlation with USNO numbers over an entire year period, especially in the extreme latitudes was very encouraging for us! It wasn't very surprising for us to find very accurate sun's elevation angle computing capability in US Naval Observatory internet site. After all navigation science is their “traditional” expertise more than any one in the US! In addition to that USNO also maintains the UTC (Universal Time Coordinated) which is the time reference of the GPS [1].
FIG. 6 shows the sun's yearly elevation angle variation from the horizon for extreme Arctic latitudes like the North Pole, the Magnetic North Pole, Barrow, Ak. and Narvik, Norway. Their GPS coordinates are given as in FIG. 6 are (90° N, 0° W), (81.3° N, 109.2° W), (71.2° N, 156.6° W) and (68.4° N, 17.4° E) respectively. As can be seen the USNO and SolarTimer results are indistinguishable over a period of a year. Another excellent source of reference and correlation is also found to be with the Australian Government Geoscience Australia, especially in the elevation angle calculations of the moon [15,16] which is also displayed in the SolarTimer compass dial.
Altitude can have fairly important effect on the sunrise, sunset times and twilight times. The magnitude of the altitude effects on the sunrise, sunset times and twilight times depend on the latitude and the date. Noon does not change as expected but the altitude effects on the twilight, sunrise and sunset times are fairly significant.
Atmospheric Effects
When talking about the sunrise and sunset issues in the extreme latitudes one has to consider the atmospheric effects as well. Atmospheric pressure and temperature are included as input variables in the software package “OEA Astronomic and Navigational Computing Utilities” which takes care of the atmospheric refraction effects on the sun's elevation angle calculations up to a certain extent for normal conditions. Since the smart phone or computer tablet does not have the temperature and atmospheric pressure information, these variables are set to some reasonable values internally. In extreme latitudes there are more dominant effects caused by Mirage's which are not taken into consideration
There is a well known phenomenon called mirage which can change the apparent sunset and sunrise under certain conditions. In certain mirage conditions it has been recorded that when the evening sun has gone down for over 20 minutes it is still clearly visible [20]. These extreme conditions generally happen when there is very high temperature gradient, higher than 2° C. per meter and generally higher than 4-5° C. per meter over the ground. These conditions occur when there is strong heating at the ground level, and an “inferior” image is commonly generated as a result of this. It is called an inferior image because the resultant image seen is under the real object, in this case the sun. Inferior images are not stable. This type of inferior mirage is very often seen on highways, deserts, airport runways and it looks like water on the surface. Hot air rises and cooler air descends. In the process, the layers mix, and turbulence will cause distortion of the image. The image might look as if it is distorted, vibrating; it might seem as if vertically extended. If there are several layers there can be multiple images. In any case, inferior images are not larger than one degree in height.
A “superior” image occurs when the air below the line of sight is colder than that above. This is called temperature inversion, since it does not represent the normal temperature gradient of the atmosphere. In this case, light rays are bent down so the image appears above the true object. Therefore they are called superior mirages. Superior mirages are in general less common than the inferior mirages, but when they occur, they tend to be more stable. This is due to the fact that the cold air has no tendency to move up or warm air to move down. Superior mirages are more common in Polar Regions, especially over large sheets of ice with a uniform low temperature, but they have been recorded in lower latitudes such as in San Francisco.
One of the most historic and striking events related to superior mirage happened to William Barents during his search for the Northeast Passage in 1596. When stuck in the ice at Novaya Zemlya, the crew saw the midwinter night come to an end with a distorted sun about two weeks earlier than expected!! The real sun was still below the horizon, but its light rays followed the curvature of the Earth. This effect is often called Novaya Zemlya. For every 111.12 km that light rays can travel parallel to the Earth's surface, the sun will appear 1° higher on the horizon. If the vertical temperature gradient is +11° C./100 m than the horizontal light rays will just follow the curvature of the Earth.
Hereafter are references that are for further reading or that are cited in this document.